Oscar.jl

Documentation	Build Status

Welcome to the OSCAR project, a visionary new computer algebra system which combines the capabilities of four cornerstone systems: GAP, Polymake, Antic and Singular.

Installation

OSCAR requires Julia 1.6 or newer. In principle it can be installed and used like any other Julia package; doing so will take a couple of minutes:

julia> using Pkg
julia> Pkg.add("Oscar")
julia> using Oscar

However, some of Oscar's components have additional requirements. For more detailed information, please consult the installation instructions on our website.

Examples of usage

julia> using Oscar
 -----    -----    -----      -      -----
|     |  |     |  |     |    | |    |     |
|     |  |        |         |   |   |     |
|     |   -----   |        |     |  |-----
|     |        |  |        |-----|  |   |
|     |  |     |  |     |  |     |  |    |
 -----    -----    -----   -     -  -     -

...combining (and extending) ANTIC, GAP, Polymake and Singular
Version 0.8.1 ...
... which comes with absolutely no warranty whatsoever
Type: '?Oscar' for more information
(c) 2019-2022 by The Oscar Development Team

julia> k, a = quadratic_field(-5)
(Imaginary quadratic field defined by x^2 + 5, sqrt(-5))

julia> zk = maximal_order(k)
Maximal order of Imaginary quadratic field defined by x^2 + 5
with basis nf_elem[1, sqrt(-5)]

julia> factorisations(zk(6))
2-element Vector{Fac{NfOrdElem}}:
 -1 * (sqrt(-5) + 1) * (sqrt(-5) - 1)
 -1 * 2 * -3

julia> Qx, x = PolynomialRing(QQ, [:x1,:x2])
(Multivariate Polynomial Ring in x1, x2 over Rational Field, fmpq_mpoly[x1, x2])

julia> R = grade(Qx, [1,2])[1]
Multivariate Polynomial Ring in x1, x2 over Rational Field graded by
  x1 -> [1]
  x2 -> [2]

julia> f = R(x[1]^2+x[2])
x1^2 + x2

julia> degree(f)
graded by [2]

julia> F = free_module(R, 1)
Free module of rank 1 over R

julia> s = sub(F, [f*F[1]])
Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as subquotient with no relations.

julia> H, mH = hom(s, quo(F, s))
(hom of (s, Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]), Map from
H to Set of all homomorphisms from Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]
represented as subquotient with no relations. to Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1] defined by a julia-function with inverse)

julia> mH(H[1])
Map with following data
Domain:
=======
s
Codomain:
=========
Subquotient of Submodule with 1 generator
1 -> e[1]
by Submodule with 1 generator
1 -> (x1^2 + x2)*e[1]

Of course, the cornerstones are also available directly:

julia> C = Polymake.polytope.cube(3);

julia> C.F_VECTOR
pm::Vector<pm::Integer>
8 12 6

julia> RP2 = Polymake.topaz.real_projective_plane();

julia> RP2.HOMOLOGY
PropertyValue wrapping pm::Array<polymake::topaz::HomologyGroup<pm::Integer>>
({} 0)
({(2 1)} 0)
({} 0)

Citing OSCAR

If you have used OSCAR in the preparation of a paper please cite it as described below:

[OSCAR]
    OSCAR -- Open Source Computer Algebra Research system, Version 0.8.1 The OSCAR Team, 2022. (https://oscar.computeralgebra.de)
[OSCAR-book]
    Christian Eder, Wolfram Decker, Claus Fieker, Max Horn, Michael Joswig, The OSCAR book, 2024.

If you are using BibTeX, you can use the following BibTeX entries:

@misc{OSCAR,
  key          = {OSCAR},
  organization = {The OSCAR Team},
  title        = {OSCAR -- Open Source Computer Algebra Research system,
                  Version 0.8.1},
  year         = {2022},
  url          = {https://oscar.computeralgebra.de},
  }

@Book{OSCAR-book,
  editor = {Eder, Christian and Decker, Wolfram and Fieker, Claus and Horn, Max and Joswig, Michael},
  title = {The OSCAR book},
  year = {2024},
}

Funding

The development of this Julia package is supported by the Deutsche Forschungsgemeinschaft DFG within the Collaborative Research Center TRR 195.